Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal $\mathfrak{gl}_2$

TL;DR

This paper constructs a $q$-deformed algebra from the quantum toroidal $rak{gl}_2$ algebra, revealing a structure combining free fermion and super Virasoro algebras, with new relations and degenerations connecting to known algebras.

Contribution

It extends the construction of $q$-deformed Virasoro algebras to the quantum toroidal $rak{gl}_2$, introducing a new algebraic structure that unifies free fermion and super Virasoro algebras.

Findings

The algebra generated by $W_i(z)$ is a $q$-deformation of $ ext{F} igoplus ext{SVir}$.

The generators admit two screening currents with limits matching those of $ ext{SVir}$.

They generate two commuting $q$-deformed Virasoro algebras that degenerate into classical Virasoro algebras.

Abstract

It is known that the $q$-deformed Virasoro algebra can be constructed from a certain representation of the quantum toroidal $\mathfrak{gl}_1$ algebra. In this paper, we apply the same construction to the quantum toroidal algebra of type $\mathfrak{gl}_2$ and study the properties of resulting generators $W_i(z)$ ($i=1,2$). The algebra generated by $W_i(z)$ can be regarded as a $q$-deformation of the direct sum $\mathsf{F} \oplus \mathsf{SVir}$, where $\mathsf{F}$ denotes the free fermion algebra and $\mathsf{SVir}$ stands for the $N=1$ super Virasoro algebra, also referred to as the $N=1$ superconformal algebra or the Neveu-Schwarz-Ramond algebra. Moreover, the generators $W_i(z)$ admit two screening currents, and we show that their degeneration limits coincide with the screening currents of $\mathsf{SVir}$. We also establish quadratic relations satisfied by $W_i(z)$ and show that they generate a pair of commuting $q$-deformed Virasoro algebras, which degenerate into two nontrivial commuting Virasoro algebras included in $\mathsf{F} \oplus \mathsf{SVir}$.